Details:
Title  Algebraicgeometric codes and multidimensional cyclic codes: a uniefied theory and algorithms for decoding using Gröbner bases  Author(s)  Chris Heegard, Keith Saints  Type  Article in Journal  Abstract  It is proved that any algebraicgeometric (AG) code can be expressed as a cross section of an extended multidimensional cyclic code. Both AG codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets: AG codes are defined over the points of an algebraic curve, and an mdimensional cyclic code is defined over the points in mdimensional space. The power of the unified theory is in its description of decoding techniques using Grobner bases. In order to fit an AG code into this theory, a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position. For curves in special position, all computations can be performed with polynomials and this also makes it possible to use the theory of Grobner bases. Next, a transform is defined for AG codes which generalizes the discrete Fourier transform. The transform is also related to a Grobner basis, and is useful in setting up the decoding problem. In the decoding problem, a key step is finding a Grobner basis for an error locator ideal. For AG codes, multidimensional cyclic codes, and indeed, any cross section of an extended multidimensional cyclic code, Sakata's algorithm can be used to find linear recursion relations which hold on the syndrome array. In this general context, the authors give a selfcontained and simplified presentation of Sakata's algorithm, and present a general framework for decoding algorithms for this family of codes, in which the use of Sakata's algorithm is supplemented by a procedure for extending the syndrome array  ISSN  00189448 
URL 
dx.doi.org/10.1109/18.476246 
Language  English  Journal  IEEE Transactions on Information Theory  Volume  41  Number  6, Part 1  Year  1995  Month  November  Translation 
No  Refereed 
No 
